Embed Size (px)
Citation preview
1
Chapter 15 – Multiple Integrals15.7 Triple Integrals
15.7 Triple Integrals
Objectives: Understand how to
calculate triple integrals Understand and apply
the use of triple integrals to different applications
15.7 Triple Integrals 2
Triple Integrals Just as we defined single integrals for
functions of one variable and double integrals for functions of two variables, so we can define triple integrals for functions of three variables.
15.7 Triple Integrals 3
Triple IntegralsLet’s first deal with the simplest case where f
is defined on a rectangular box:
, , , ,B x y z a x b c y d r z s
15.7 Triple Integrals 4
Triple IntegralsThe first step is
to divide B into sub-boxes—by dividing:
◦ The interval [a, b] into l subintervals [xi-1, xi] of equal width Δx.
◦ [c, d] into m subintervals of width Δy.◦ [r, s] into n subintervals of width Δz.
15.7 Triple Integrals 5
Triple IntegralsThe planes through
the endpoints of these subintervals parallel to the coordinate planes divide the box B into lmn sub-boxes
◦ Each sub-box has volume ΔV = Δx Δy Δz
1 1 1, , ,ijk i i j j k kB x x y y z z
15.7 Triple Integrals 6
Triple IntegralsThen, we form the triple Riemann sum
where the sample point is in Bijk.
* * *
1 1 1
, ,l m n
ijk ijk ijki j k
f x y z V
* * *, ,ijk ijk ijkx y z
15.7 Triple Integrals 7
Triple IntegralsThe triple integral of f over the box B is:
if this limit exists.◦ Again, the triple integral always exists if f
is continuous.
* * *
, ,1 1 1
, , lim , ,l m n
ijk ijk ijkl m n
i j kB
f x y z dV f x y z V
15.7 Triple Integrals 8
Fubini’s Theorem for Triple Integrals Just as for double integrals, the practical
method for evaluating triple integrals is to express them as iterated integrals, as follows.
If f is continuous on the rectangular box B = [a, b] x [c, d] x [r, s], then
, , , ,s d b
r c aB
f x y z dV f x y z dx dy dz
15.7 Triple Integrals 9
Fubini’s Theorem
The iterated integral on the right side of Fubini’s Theorem means that we integrate in the following order:
1. With respect to x (keeping y and z fixed)2. With respect to y (keeping z fixed)3. With respect to z
, , , ,s d b
r c aB
f x y z dV f x y z dx dy dz
15.7 Triple Integrals 10
Example 1 – pg. 998 # 4Evaluate the triple integral.
1 2
0 0
2yx
x
xyz dz dy dx
15.7 Triple Integrals 11
Integral over a Bounded Region
We restrict our attention to:
◦Continuous functions f
◦Certain simple types of regions
15.7 Triple Integrals 12
Type I Region Eq.5A solid region E is said to be of type 1 if it
lies between the graphs of two continuous functions of x and y. That is,
where D is the projection of E onto the xy-plane.
1 2, , , , , ,E x y z x y D u x y z u x y
15.7 Triple Integrals 13
Type I RegionNotice that:
◦ The upper boundary of the solid E is the surface with equation z = u2(x, y).
◦ The lower boundary is the surface z = u1(x, y).
15.7 Triple Integrals 14
Type I Region Eq. 6 If E is a type 1 region given by Equation 5,
then we have Equation 6:
2
1
,
,, , , ,
u x y
u x yE D
f x y z dV f x y z dz dA
15.7 Triple Integrals 15
Type II RegionA solid region E is said to be of type 1 if it
lies between the graphs of two continuous functions of x and y. That is,
where D is the projection of E onto the yz-plane.
1 2, , , , , ,E x y z y z D u y z x u y z
15.7 Triple Integrals 16
Type II RegionNotice that:
◦ The back surface is x = u1(y, z).
◦ The front surface is x = u2(y, z).
15.7 Triple Integrals 17
Type II Region Eq. 10For this type of region we have:
2
1
,
,, , , ,
u y z
u y zE D
f x y z dV f x y z dx dA
15.7 Triple Integrals 18
Type III RegionFinally, a type 3 region is of the form
where:◦ D is the projection of E
onto the xz-plane.
1 2, , , , ( , ) ,E x y z x z D u x z y u x z
15.7 Triple Integrals 19
Type III RegionNotice that:
◦ y = u1(x, z) is the left surface.
◦ y = u2(x, z) is the right surface.
15.7 Triple Integrals 20
Type III Region Eq. 11For this type of region, we have:
2
1
,
,, , , ,
u x z
u x zE D
f x y z dV f x y z dy dA
15.7 Triple Integrals 21
VisualizationRegions of Integration in Triple Int
egrals
15.7 Triple Integrals 22
Example 2 – pg. 998 # 10Evaluate the triple integral.
5cos , where
, , |0 1, 0 , 2
E
yz x dV
E x y z x y x x z x
15.7 Triple Integrals 23
Example 3 – pg. 998 # 20Use a triple integral to find the
volume of the given solid.
2
The solid bounded by the cylinder
and the planes 0, 4,
and 9.
y x z z
y
15.7 Triple Integrals 24
Example 4 – pg. 998 # 22Use a triple integral to find the
volume of the given solid.
2 2
The solid enclosed by the paraboloid
+ and the plane 16.x y z x
15.7 Triple Integrals 25
Example 5 – pg. 999 # 36Write five other iterated integrals
that are equal to the given iterated integral.
21
0 0 0
, ,yx
f x y z dz dy dx
15.7 Triple Integrals 26
More Examples
The video examples below are from section 15.7 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦Example 1◦Example 3◦Example 6
